Extensions of Buchi's Higher Powers Problem to Positive Characteristic

Abstract

Büchi's n-th power problem on Q asks whether there exist an integer M such that the only monic polynomials F ∈ Q[X] of degree n satisfying that F (1),. .. , F (M) are n-th power rational numbers, are precisely of the form F (X) = (X +c) n for some c ∈ Q. In this paper, we study analogues of this problem for algebraic function fields of positive characteristic. We formulate and prove an analogue (indeed, such a formulation for n > 2 was missing in the literature due to some unexpected phenomena), which we use to derive some definability and undecidability consequences. Moreover, in the case of characteristic zero we extend some known results by improving the bounds for M (from quadratic on n to linear on n). Contents Büchi's undecidability problem. Does there exists an algorithm for the following decision problem? Given diagonal quadratic forms Q j (x 1 ,. .. , x n) = a j1 x 2 1 +. .. + a jn x 2 n (for j = 1,. .. , m) with integer coefficients, and given b = (b j) j ∈ Z m , decide whether or not there is a tuple of integers a ∈ Z n such that for each j = 1,. .. , m we have Q j (a) = b j. More precisely, Büchi formulated an arithmetic problem which, if true, would imply that the previous problem has a negative answer. See [6, 8, 9] for more details. The arithmetic problem formulated by Büchi is the following.

Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials

“…Analogues of Question 1.1 have been considered for most classical rings (but in the case of number fields, under some well-known conjectures, like Bombieri-Lang for surfaces, or some version of ABC). Relevant results in positive characteristic can be found in [Pa11] (the analogue of Büchi's problem for any power over fields with a prime number of elements), and in [PaW15] (over rings of functions), who generalize previous results in [PhV06,PhV10,ShV10,AW11,AHW13]. For a general survey on Büchi's problem and its extensions to other structures and higher powers, see [PaPhV10].…”

Uniform positive existential interpretation of the integers in rings of entire functions of positive characteristic

“…Let us recall the following theorem from [17] (see also [16], [20] and [13]). It gives a solution to Büchi's n squares problem for polynomial rings in positive characteristic.…”

On Pisot's $d$-th root conjecture for function fields and related GCD estimates

“…Our proof of Theorem 2 contains two major ingredients, both rely on the special features of function fields of characteristic zero. One of the ingredients is the result (restated as Theorem 23 in Section 4) by Pasten and Wang [12] motivated by Büchi's d-th power problem, which has a similar flavor as Pisot's d-th root conjecture but arising from different purposes. While working on an undecidability problem related to Hilbert's tenth problem in the 1970s, Büchi formulated a related arithmetic problem, which can be stated in more generality as follows: Let k be a number field.…”

“…In particular, the analog over function fields of characteristic zero was solved completely, even with an explicit bound on M ; see [2] and [11]. We refer to [12] for a survey of relevant works. The other ingredient, which is also developed in this paper, is the function-field analog of the recent work of Levin [7] for number fields and Levin-Wang [8] for meromorphic functions on GCD estimates of two multivariable polynomials over function fields evaluated at arguments which are S-units.…”

“…. , x n ]) obtained from the same construction of h with K replaced by L; similar for the notation hL , and O L, S , N L, S , N L, S , where S ⊂ C L (k) is a finite subset, and C L be a smooth projective curve over k such that L = k(C L ).We need the following result from[12, Proposition 2.4].Proposition 21. Let α be a nonconstant algebraic element over K with [K(α) : K] = m. Denote by L = K(α) and let C L be a smooth projective curve over k of genus g L such that L = k(C L ).…”